\(\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx\) [602]

Optimal result

Integrand size = 29, antiderivative size = 228 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {13 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \]

[Out]

13/256*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-2/9*a^2*cot(d*x+c)^9/d+13/256*a^2*cot(d*x+c)*csc(d*x+c
)/d-9/128*a^2*cot(d*x+c)*csc(d*x+c)^3/d+5/48*a^2*cot(d*x+c)^3*csc(d*x+c)^3/d-1/8*a^2*cot(d*x+c)^5*csc(d*x+c)^3
/d-1/32*a^2*cot(d*x+c)*csc(d*x+c)^5/d+1/16*a^2*cot(d*x+c)^3*csc(d*x+c)^5/d-1/10*a^2*cot(d*x+c)^5*csc(d*x+c)^5/
d

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2691, 3853, 3855, 2687, 14} \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {13 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]

[Out]

(13*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) + (13*a^2
*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^2*Cot[c + d*x]^3*Csc[
c + d*x]^3)/(48*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a
^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cot ^6(c+d x) \csc ^3(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^4(c+d x)+a^2 \cot ^6(c+d x) \csc ^5(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx \\ & = -\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{2} a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx-\frac {1}{8} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {1}{16} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {1}{16} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{32} a^2 \int \csc ^5(c+d x) \, dx-\frac {1}{64} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{128} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{128} \left (5 a^2\right ) \int \csc (c+d x) \, dx \\ & = \frac {5 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{256} \left (3 a^2\right ) \int \csc (c+d x) \, dx \\ & = \frac {13 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.67 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.55 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^{10}(c+d x) \left (2732940 \cos (c+d x)+1151640 \cos (3 (c+d x))+388248 \cos (5 (c+d x))-135870 \cos (7 (c+d x))-8190 \cos (9 (c+d x))-515970 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+859950 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-491400 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+184275 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-40950 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4095 \cos (10 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+515970 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-859950 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+491400 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-184275 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+40950 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4095 \cos (10 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1075200 \sin (2 (c+d x))+1044480 \sin (4 (c+d x))+414720 \sin (6 (c+d x))+51200 \sin (8 (c+d x))-5120 \sin (10 (c+d x))\right )}{41287680 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/41287680*(a^2*Csc[c + d*x]^10*(2732940*Cos[c + d*x] + 1151640*Cos[3*(c + d*x)] + 388248*Cos[5*(c + d*x)] -
135870*Cos[7*(c + d*x)] - 8190*Cos[9*(c + d*x)] - 515970*Log[Cos[(c + d*x)/2]] + 859950*Cos[2*(c + d*x)]*Log[C
os[(c + d*x)/2]] - 491400*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 184275*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2
]] - 40950*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 4095*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 515970*Log[
Sin[(c + d*x)/2]] - 859950*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 491400*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/
2]] - 184275*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 40950*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 4095*Cos[
10*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 1075200*Sin[2*(c + d*x)] + 1044480*Sin[4*(c + d*x)] + 414720*Sin[6*(c +
d*x)] + 51200*Sin[8*(c + d*x)] - 5120*Sin[10*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.09

method	result	size
parallelrisch	\(-\frac {\left (520 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {40 \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {5 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {120 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {95 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+60 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {320 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+90 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-240 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (-240+60 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {95 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {320 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {120 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+90 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {40 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{10240 d}\)	\(248\)
risch	\(-\frac {a^{2} \left (4095 \,{\mathrm e}^{19 i \left (d x +c \right )}+67935 \,{\mathrm e}^{17 i \left (d x +c \right )}-194124 \,{\mathrm e}^{15 i \left (d x +c \right )}+215040 i {\mathrm e}^{14 i \left (d x +c \right )}-575820 \,{\mathrm e}^{13 i \left (d x +c \right )}+322560 i {\mathrm e}^{16 i \left (d x +c \right )}-1366470 \,{\mathrm e}^{11 i \left (d x +c \right )}-829440 i {\mathrm e}^{6 i \left (d x +c \right )}-1366470 \,{\mathrm e}^{9 i \left (d x +c \right )}+1075200 i {\mathrm e}^{12 i \left (d x +c \right )}-575820 \,{\mathrm e}^{7 i \left (d x +c \right )}-194124 \,{\mathrm e}^{5 i \left (d x +c \right )}-645120 i {\mathrm e}^{10 i \left (d x +c \right )}+67935 \,{\mathrm e}^{3 i \left (d x +c \right )}-92160 i {\mathrm e}^{4 i \left (d x +c \right )}+4095 \,{\mathrm e}^{i \left (d x +c \right )}-51200 i {\mathrm e}^{2 i \left (d x +c \right )}+5120 i\right )}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}+\frac {13 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}-\frac {13 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}\)	\(260\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(312\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(312\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/10240*(520*ln(tan(1/2*d*x+1/2*c))+cot(1/2*d*x+1/2*c)^10+40/9*cot(1/2*d*x+1/2*c)^9+5/2*cot(1/2*d*x+1/2*c)^8-
120/7*cot(1/2*d*x+1/2*c)^7-95/3*cot(1/2*d*x+1/2*c)^6+60*cot(1/2*d*x+1/2*c)^4+320/3*cot(1/2*d*x+1/2*c)^3+90*cot
(1/2*d*x+1/2*c)^2-240*cot(1/2*d*x+1/2*c)-(-240+60*tan(1/2*d*x+1/2*c)^3-95/3*tan(1/2*d*x+1/2*c)^5+320/3*tan(1/2
*d*x+1/2*c)^2-120/7*tan(1/2*d*x+1/2*c)^6+90*tan(1/2*d*x+1/2*c)+tan(1/2*d*x+1/2*c)^9+5/2*tan(1/2*d*x+1/2*c)^7+4
0/9*tan(1/2*d*x+1/2*c)^8)*tan(1/2*d*x+1/2*c))*a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {8190 \, a^{2} \cos \left (d x + c\right )^{9} + 15540 \, a^{2} \cos \left (d x + c\right )^{7} - 69888 \, a^{2} \cos \left (d x + c\right )^{5} + 38220 \, a^{2} \cos \left (d x + c\right )^{3} - 8190 \, a^{2} \cos \left (d x + c\right ) - 4095 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4095 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5120 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{9} - 9 \, a^{2} \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/161280*(8190*a^2*cos(d*x + c)^9 + 15540*a^2*cos(d*x + c)^7 - 69888*a^2*cos(d*x + c)^5 + 38220*a^2*cos(d*x +
c)^3 - 8190*a^2*cos(d*x + c) - 4095*(a^2*cos(d*x + c)^10 - 5*a^2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^6 - 10*
a^2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2) + 4095*(a^2*cos(d*x + c)^10 - 5*a
^2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^6 - 10*a^2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d
*x + c) + 1/2) + 5120*(2*a^2*cos(d*x + c)^9 - 9*a^2*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^10 - 5*d*cos
(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**11*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.20 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {63 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {5120 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/161280*(63*a^2*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(
d*x + c))/(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1)
- 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 210*a^2*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 5
5*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 +
1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 5120*(9*tan(d*x + c)^2 + 7)*a^2/tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.42 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {126 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3990 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 13440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 65520 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {191906 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 13440 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3990 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1290240*(126*a^2*tan(1/2*d*x + 1/2*c)^10 + 560*a^2*tan(1/2*d*x + 1/2*c)^9 + 315*a^2*tan(1/2*d*x + 1/2*c)^8 -
2160*a^2*tan(1/2*d*x + 1/2*c)^7 - 3990*a^2*tan(1/2*d*x + 1/2*c)^6 + 7560*a^2*tan(1/2*d*x + 1/2*c)^4 + 13440*a
^2*tan(1/2*d*x + 1/2*c)^3 + 11340*a^2*tan(1/2*d*x + 1/2*c)^2 - 65520*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 3024
0*a^2*tan(1/2*d*x + 1/2*c) + (191906*a^2*tan(1/2*d*x + 1/2*c)^10 + 30240*a^2*tan(1/2*d*x + 1/2*c)^9 - 11340*a^
2*tan(1/2*d*x + 1/2*c)^8 - 13440*a^2*tan(1/2*d*x + 1/2*c)^7 - 7560*a^2*tan(1/2*d*x + 1/2*c)^6 + 3990*a^2*tan(1
/2*d*x + 1/2*c)^4 + 2160*a^2*tan(1/2*d*x + 1/2*c)^3 - 315*a^2*tan(1/2*d*x + 1/2*c)^2 - 560*a^2*tan(1/2*d*x + 1
/2*c) - 126*a^2)/tan(1/2*d*x + 1/2*c)^10)/d

Mupad [B] (verification not implemented)

Time = 11.30 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.57 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {19\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {9\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {19\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6144\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {13\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}+\frac {3\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^11,x)

[Out]

(19*a^2*cot(c/2 + (d*x)/2)^6)/(6144*d) - (a^2*cot(c/2 + (d*x)/2)^3)/(96*d) - (3*a^2*cot(c/2 + (d*x)/2)^4)/(512
*d) - (9*a^2*cot(c/2 + (d*x)/2)^2)/(1024*d) + (3*a^2*cot(c/2 + (d*x)/2)^7)/(1792*d) - (a^2*cot(c/2 + (d*x)/2)^
8)/(4096*d) - (a^2*cot(c/2 + (d*x)/2)^9)/(2304*d) - (a^2*cot(c/2 + (d*x)/2)^10)/(10240*d) + (9*a^2*tan(c/2 + (
d*x)/2)^2)/(1024*d) + (a^2*tan(c/2 + (d*x)/2)^3)/(96*d) + (3*a^2*tan(c/2 + (d*x)/2)^4)/(512*d) - (19*a^2*tan(c
/2 + (d*x)/2)^6)/(6144*d) - (3*a^2*tan(c/2 + (d*x)/2)^7)/(1792*d) + (a^2*tan(c/2 + (d*x)/2)^8)/(4096*d) + (a^2
*tan(c/2 + (d*x)/2)^9)/(2304*d) + (a^2*tan(c/2 + (d*x)/2)^10)/(10240*d) - (13*a^2*log(tan(c/2 + (d*x)/2)))/(25
6*d) + (3*a^2*cot(c/2 + (d*x)/2))/(128*d) - (3*a^2*tan(c/2 + (d*x)/2))/(128*d)